Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Interfaces between
Probability and Geometry
(Prospects in Mathematics
Durham, 15 December 2007)
Wilfrid Kendall
w.s.kendall@warwick.ac.uk
Department of Statistics, University of Warwick
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Introduction
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Brownian motion
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Brownian motion (I)
“Infinitesimal random walk” relates to numerical
analysis: Courant, Friedrichs, and Lewy 1928.
f (x) =
1
(f (x + n) + f (x + e) + f (x + s) + f (x + w))
4
converging in the limit to
∆f = 0
f (x) = E [f (BT )|B0 = x]
(martingales, stochastic calculus, . . . )
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
Brownian motion (II)
From random walk to Brownian motion and beyond:
extends idea of Central Limit Theorem;
stochastic modelling;
complex analysis via Lévy’s Theorem;
stochastic Itô calculus (finance!).
Explicit representation f (x) = E [f (BT )|B0 = x] useful:
in case of irregular boundaries;
when generalizing to diffusions;
as part of general approach to random processes on
fractals.
Significant generalizations:
from stochastic differential equations (SDE) to stochastic
partial differential equations (SPDE);
measure-valued diffusions (let BM reproduce and die);
nonlinear elliptic variational problems (need to generalize
martingales, expectations).
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Brownian motion (III)
Neumann heat kernel
Reflecting Brownian motion:
prevent all moves outside of the domain D;
related to heat flow in insulated domain;
probability (heat) kernel ptD (x, y ) is probability density of Bt
at Y if B is begun at x.
QUESTION: does ptD (x, y ) depend monotonely on D?
yes at large times;
not necessarily at very short times;
yes for convex well-separated domains
(WSK 1989);
no for general convex domains
(Bass and Burdzy 1993).
(“fly in convex room” example)
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Brownian motion (IV)
Stochastic Loewner Evolution (SLE)
Consider H the upper half-plane, and the differential
equation
∂
2
√
gt (z) =
∂t
gt (z) − κBt
defined for Brownian motion B begun at 0, with
g0 (z) = z ∈ H.
This fails to be defined after time
Tz = sup{t : |gs (z) − κBs | > 0 for s ≤ t}.
Then Kt = {z ∈ H : Tz ≤ t} is (chordal) Stochastic
Loewner Evolution SLEκ , closely linked to many stochastic
boundaries, the study of which won a Fields medal for
Wendelin Werner in 2006.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Brownian motion: preparations and directions
Preparations:
measure theory, measure-theoretic probability, analysis;
Øksendal (2003) (book on SDE);
Revuz and Yor (1999) (book on martingales and Brownian
motion).
(Random) directions:
What might be the statistical consequences of domain
non-monotonicity of the Neumann heat kernel?
How might the theory of SLE be used in applied probability
to model interfaces?
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Percolation
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Percolation (I)
Consider bond percolation: pipe segments are independently
open with probability p, closed with probability 1 − p. As p
increases, when does an infinite open cluster first appear?
History: Broadbent and
Hammersley (1957)
Clearly a critical pH should exist.
Theorem (Kesten, 1980):
pH = 1/2.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Percolation (II)
There are many other kinds of percolation
site percolation;
triangular lattice site percolation
(Cardy’s formula in critical case, SLE6 );
Poisson-Voronoi site percolation
(Bollobas and Riordan 2006, pH = 1/2);
Percolation on Cayley graphs of groups . . .
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Percolation (III)
Percolation and non-amenability
Grimmett and Newman (1990) study bond-percolation in
T × Zd where T is a regular tree. (Hard to visualize!)
Tree-bonds have different probabilities from space-bonds.
They show there are at least two phase transitions:
0-to-many-infinite-clusters, then many-to-1-infinite-cluster.
For Cayley graphs, existence of two phase transitions is
related to whether the underlying group is non-amenable
(cannot possess an invariant mean).
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Percolation (IV)
For particular tree-like graphs
arising in quad-tree-based image
analysis, WSK and Wilson (2003)
show two phase transitions using
methods related to hyperbolicity.
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Percolation: preparations and directions
Preparations:
first-year undergraduate probability(!) and plenty of
mathematical analysis;
Grimmett (1999) (book on percolation, mostly Euclidean);
Benjamini and Schramm (1996) “Percolation beyond Zd ,
many questions and a few answers”.
(Random) directions:
Investigate whether the Bollobas and Riordan (2006) result
has consequences for spatial epidemics.
Can uniqueness-of-infinite-cluster transition be related to
notions of metric-space hyperbolicity for graphs?
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
The Ising model
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
The Ising model (I)
Each node is spin-up (+, σ = +1) or
spin-down (−, σ = −1);
Probability of configuration
proportional to:





X
X
1 




exp −
−J
σi σj
= exp β
σi σj  .
kT
ij
ij
Phase transition for infinite lattice:
computable in dimension 2;
Local specification versus model on
infinite grid;
Image analysis: superimpose a
second fixed/conditioned lattice,
linked by different β.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
The Ising model (II)
Image analysis with Ising models:
“heat-bath” algorithm (MCMC);
Coupling from the Past (CFTP);
Good image reconstruction:
low “temperature” plus
strong influence of image;
Recent theoretical evidence suggests fast
convergence (of modified algorithm);
More sophisticated approaches: use
“quad-trees” (superimposed coarser grids).
Issue: identify phase transitions;
Use the Fortuin-Kastelyn inequalities to
estimate using percolation!
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Ising model: preparations and directions
Preparations:
once again, first-year undergraduate probability(!) and
plenty of mathematical analysis;
Kindermann and Snell (1980) (excellent introduction, freely
available on web!);
(Random) directions:
The quad-tree work is now focussed on getting an
understanding of when interfaces propagate.
SLE is conjectured to be related to the behaviour of the
Ising model at criticality.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
Networks and stochastic geometry
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
Networks and stochastic geometry (I)
What is the shortest network to connect up
the red sites using the grid?
The answer is the rectilinear Steiner Tree.
In general, computation of this is an
NP-complete algorithm!
The Euclidean variant is also NP-complete
of course.
Steiner trees are good for minimal length
connection, but bad for efficient
connections. How much more expensive to
do better?
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Networks and stochastic geometry (II)
Aldous and WSK 2008: take √
the Euclidean Steiner tree for
n locations in a rectangle [0, n]2 ;
total Steiner tree network length is of order n;
we can reduce average connection length to within about
O(log n) of best possible, as follows:
add a very small number of extra random lines to generate
efficient long-range connections;
add a small amount of extra connectivity to ensure one can
move efficiently onto the lines using the Steiner tree;
total extra network length is just o(n)!
Random line: choose uniform
random orientation θ, choose
uniform random (signed) distance r
from reference point.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
Conclusion
References
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Conclusion (continued)
There is plenty to do in probability theory, whether you want:
deep and difficult theories with powerful links to other
areas of mathematics;
stochastic techniques and concepts which underly the
finance industry;
or pretty and challenging applied problems with surprising
results.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Bibliography
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added where known. Access to such URLs is not universal: in case of difficulty you
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Aldous, D. J. and WSK (2008).
Short-length routes in low-cost networks via Poisson line patterns.
Advances in Applied Probability 40(1), to appear, , and
http://arxiv.org/abs/math.PR/0701140
.
Bass, R. F. and K. Burdzy (1993).
On domain monotonicity of the Neumann heat kernel.
J. Funct. Anal. 116(1), 215–224.
Benjamini, I. and O. Schramm (1996).
Percolation beyond Zd , many questions and a few answers.
Electron. Comm. Probab. 1, no. 8, 71–82 (electronic).
Bollobas, B. and O. Riordan (2006).
The critical probability for random voronoi percolation in the plane is 1/2.
Probability Theory and Related Fields 136, 417.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Broadbent, S. R. and J. M. Hammersley (1957).
Percolation processes. I. Crystals and mazes.
Proc. Cambridge Philos. Soc. 53, 629–641.
Courant, R., K. Friedrichs, and H. Lewy (1928).
Über die partiellen Differenzengleichungen der mathematischen Physik.
Math. Ann. 100(1), 32–74.
Grimmett, G. (1999).
Percolation (Second ed.), Volume 321 of Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences].
Berlin: Springer-Verlag.
Grimmett, G. R. and C. M. Newman (1990).
Percolation in ∞ + 1 dimensions.
In Disorder in physical systems, Oxford Sci. Publ., pp. 167–190. New York: Oxford
Univ. Press.
Kindermann, R. and J. L. Snell (1980).
Markov random fields and their applications, Volume 1 of Contemporary
Mathematics.
Providence, R.I.: American Mathematical Society.
Introduction
Brownian motion
Percolation
Ising model
Stochastic geometry
Conclusion
References
Øksendal, B. (2003).
Stochastic differential equations (Sixth ed.).
Universitext. Berlin: Springer-Verlag.
An introduction with applications.
Revuz, D. and M. Yor (1999).
Continuous martingales and Brownian motion (Third ed.), Volume 293 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences].
Berlin: Springer-Verlag.
WSK (1989).
Coupled Brownian motions and partial domain monotonicity for the Neumann heat
kernel.
.
Journal of Functional Analysis 86, 226–236,
WSK and R. G. Wilson (2003, March).
Ising models and multiresolution quad-trees.
; Also University of Warwick
Advances in Applied Probability 35(1), 96–122,
Department of Statistics Research Report 402 .